Unconventional superconductivity in topological Kramers nodal-line semimetals

Crystalline symmetry is a defining factor of the electronic band topology in solids, where many-body interactions often induce a spontaneous breaking of symmetry. Superconductors lacking an inversion center are among the best systems to study such effects or even to achieve topological superconductivity. Here, we demonstrate that TRuSi materials (with T a transition metal) belong to this class. Their bulk normal states behave as three-dimensional Kramers nodal-line semimetals, characterized by large antisymmetric spin-orbit couplings and by hourglass-like dispersions. Our muon-spin spectroscopy measurements show that certain TRuSi compounds spontaneously break the time-reversal symmetry at the superconducting transition, while unexpectedly showing a fully gapped superconductivity. Their unconventional behavior is consistent with a unitary (s + ip) pairing, reflecting a mixture of spin singlets and spin triplets. By combining an intrinsic time-reversal symmetry-breaking superconductivity with nontrivial electronic bands, TRuSi compounds provide an ideal platform for investigating the rich interplay between unconventional superconductivity and the exotic properties of Kramers nodal-line/hourglass fermions.

Supplementary Materials to "Unconventional superconductivity in topological Kramers nodal-line semimetals" Figure S1 (a) Temperature-dependent electrical resistivity of TRuSi (T = Ti, Nb, Ta, and Hf) from 300 K down to 2 K in the absence of a magnetic field. (b) The enlarged plot shows the resistivity of NbRuSi and TaRuSi below 10 K. Note that, in HfRuSi, the transition at 8.8 K might be due to a minority superconducting phase (i.e., HfRu), as confirmed by the magnetic susceptibility data in Figure S2 below.

Figure S2
Temperature-dependent magnetic susceptibility (a) and specific heat (b) of HfRuSi. The susceptibility was measured in an applied field of 1 mT using the ZFC and FC protocols. Note that the superconducting volume fraction is only a marginal 2%, as confirmed by the absence of any anomaly in the specific-heat data at 8.8 K.

Figure S3
Temperature-dependent heat capacity of TRuSi (T = Ti, Nb, Ta, and Hf), measured in zero field from 2 to 300 K. The solid lines represent fits to a combined Debye-and Einstein model, with the dash-dotted-and dashed lines referring to the two components. The estimated Debye-and Einstein temperatures are listed in the respective panels. The resulting electronic specific-heat coefficients are n = 3. 1, 9.4, 8.4, and 2.9 mJ/mol-K 2 for TiRuSi, NbRuSi, TaRuSi and HfRuSi, respectively.

Figure S4
Temperature-dependent specific heat C(T)/T of NbRuSi and TaRuSi at low temperature. Note the clear jump at the respective superconducting transitions.

Figure S5
Nuclear-moment-related Lorentzian relaxation rates ZF derived from the ZF-SR data vs. reduced temperature T/Tc for NbRuSi (a) and TaRuSi (b), respectively. In either case, ZF(T) does not show anomalies across Tc, although ZF(T) (see main text) shows a weak but clear increase in the superconducting state. The dashed lines mark the average value of ZF.

Figure S6
Real part of the Fast Fourier transforms of the TF-µSR data of TaRuSi, collected at 0.3 K (a) and 6 K (b), and shown in Figure 3(b) of the main text. Solid lines are fits to Eq.
(1) in the main text using two oscillations, here shown also separately as dash-dotted lines, together with a background contribution; while the black dashed line represents a fit to Eq.
(1) with a single oscillation, clearly in poor agreement with the experimental data. NbRuSi exhibits similar features.

Figure S7
Superfluid density vs. reduced temperature T/Tc for NbRuSi (a) and TaRuSi (b). The solid lines are fits to the single-gap s-, p-, d-, and symmetric (s+ip)-wave models, while the dash-dotted lines represent fits to the (s+s)-wave model with two gaps. Here the s-wave model is identical to the (s+ip) model without considering SOC in the main text. It is noted that the (s+ip) model in the main text is non-symmetric.     Similar to TRuSi, presented in the manuscript, all these materials adopt the same TiFeSitype crystal structure (Ima2, No. 46). The KWP are marked by orange circles, while the KNL are presented by blue-(along Γ-Z) or green lines (along R-W), respectively. Clearly, the KWP and KNL can be shifted to EF either by electron doping (e.g., deposition of potassium) or by chemical substitution. In certain cases, the Kramers Weyl points are already located at EF [see, e.g., the T point in panels (a), (e), and (f)].

Figure S13
Illustration of hourglass-shaped dispersion of (a) TaRuSi (doped with 1.5 electrons per cell) and (b) TiIrGe along the -R-Z lines. In both cases, the hourglass dispersion crosses the Fermi level EF located at 0 eV.  Note S1 In the main text, we use a simplified ⋅ -type Hamiltonian, given by: where ( ) = 2 + 2 + 2 , and ⃗( ) = ( 1 , 2 , 0) is the Rashba-type spinorbital coupling. Here, the Pauli matrix ⃗ denotes the spin degree of freedom, and , , , , 1 , 2 are free parameters that depend on the material details. We note that higher order 3 terms, as e.g., , are ignored, since along the − direction the SOC is rather weak, and it can be ignored close to the Fermi energy (see Figs. 4B and 4E in the main text). In the case of a clean superconducting system, we find that the singlet-triplet 1 + 2 unitary pairing is compatible with the NbRuSi and TaRuSi superconductivity. Here the 1 irreducible representation (irrep) accounts for the spin-singlet channel (Δ ) and the 2 irrep represents the spin-triplet channel (Δ ). Thus, we label this pairing symmetry 1 + 2 as nonsymmetric (s+ip) pairing, because it naturally breaks the crystal symmetry, i.e., by reducing the 2 point group down to the 2 point group. However, it is well known that the non-1 irrep spin-triplet d-vector would be suppressed by the presence of SOC. Namely, the spin-triplet pairing strength Δ is suppressed by the SOC, which in turn increases the superconducting free energy. Nevertheless, such non-symmetric (s+ip) pairing can still be stabilized by the formation of superconductivity-induced spin magnetism.
The superconductivity-induced internal spin magnetization in the broken-TRS state can be understood based on the standard Ginzburg-Landau theory. The symmetry-allowed Ginzburg-Landau free energy for a homogeneous superconductor is given by: = ( )|Δ | 2 + ( )|Δ | 2 + | ⃗⃗⃗ | 2 + 1 (Δ Δ ) 2 + 2 Δ Δ + . ., where Δ , Δ , ⃗⃗⃗ are the order parameters for spin-singlet pairing, spin-triplet pairing, and spinmagnetization, respectively. Here, the high-order terms are not shown. The ( ) and ( ) determine the superconducting transition temperatures and are assumed to be close in two channels due to the absence of inversion symmetry. There is no spontaneous ferromagnetic ordering in the currently studied materials, therefore, we can simply set > 0. Here, 1 ≠ 0 gives rise to the Δ + Δ pairing that spontaneously breaks the TRS, and the 2 term determines the superconductivity-induced internal magnetic field ⃗⃗⃗ (spin magnetization) by the symmetry constraint 2 = 1 ⊗ 2 , which can be readily detected by the ZF-SR technique. The minimization of free energy leads to: Thus, the superconducting pairings can be expressed as:

Δ( ) = [Δ + Δ ( 1 , 2 , 3 ) ⋅ ⃗]( ).
By definition of unitary pairing, Δ , Δ , and the spin-triplet ⃗ ( ) vector are all real. The relative phase between Δ and Δ ( ± = ± ) leads to the spontaneous breaking of time-reversal symmetry. Such pairings were used to analyze the superconducting superfluid density after projecting to the Fermi surfaces upon assuming a weak-pairing limit. Precisely, this energy loss is due to the suppression of by the spin-orbit coupling , while the energy gain is given by − 2 2 | * | 2 . Both these two effects are closely related to , the fully-gapped 1 + 2 pairing becomes possible if the spin magnetism term wins (i.e., the energy gain is larger than the energy loss). In real materials, this depends crucially on the detailed parameters (e.g., , , etc.).
In a TRS-breaking scenario for a clean system with weak SOC, we also discuss how to pairing is the spontaneous breaking of glide symmetry which, in turn, can wipe out the hourglass fermions close to the Fermi energy. Such effects can be used to identify the pairing symmetries.

Note S2
According to two previous theoretical works (Refs. [71,72]  ] ≤ 0 with = | ⃗( )|/ . Here 〈⋯ 〉 represents the integration over the Fermi surface. It proves that only for ⃗ ∥ ⃗ at any momenta, is unaffected by SOC. Therefore, an 1 irrep spin-triplet pairing parallel to the SOC-vector is the most favorable one, i.e., it minimizes the free energy in the superconducting state. As for the other spin-triplet pairings, we usually have | ⃗ ( )| 2 − | ⃗( ) ⋅ ⃗ ( )| 2 ≥ 0, which means that the superconducting transition temperature is reduced by the presence of SOC, and thus, the pairing strength Δ is reduced. As a result, the superconducting free energy increases. However, the direct application of these assumptions to superconductors that spontaneously break TRS is questionable, as currently, the theory has not fully answered this issue. In other words, the TRS-breaking pairing symmetry represents one of the new pairing mechanisms that can coexist with orbital-or spin magnetism, and this is beyond the above theoretical works.
We now discuss the TRS-breaking pairing used in the manuscript. The non-symmetric (s+ip) pairing can describe the superfluid density quite well (see Fig. 3 in the main text). There are two main reasons for the application of this non-symmetric (s+ip) pairing to the (Nb,Ta)RuSi superconductors: (1) Anisotropic SOC The compounds are anisotropic due to their low crystal symmetry ( 2 ). Both the effective masses as well as the SOC are different along the x-and y-directions. As discussed in the above Note S1 for the single-band the effective ⋅ Hamiltonian is 0 ( ) = ( ) 0 + ⃗( ) ⋅ ⃗, where ( ) = 2 + 2 + 2 and ⃗( ) = ( 1 , 2 , 0). To show the anisotropic features on Fermi surfaces, we rescale the momentum and the coefficients for SOC, Thus, the Hamiltonian becomes 0 ( ) = ( 2 + 2 + ( )) 0 + ( 1 + 2 ), where = 2 − , with the Fermi energy. Then, the anisotropic SOC is reflected by the difference between 1 and 2 . Below, we discuss the effect of anisotropic SOC on the spintriplet -vector.
The 2 irrep spin-triplet is represented by the -vector ( ) = ( 1 , 2 , 3 ). According to previous theoretical work (see Ref. [71] in the main text), the 3 term should vanish. We calculated the superconducting transition temperature / in the spin-triplet channel.
Without loss of generality, we set 1 > 2 . And the numerical results are shown in Fig. S14 above. In Fig. S14 (a), the inset shows the two Fermi surfaces in the -space with = 0 for 1 = 1 and 2 = 0.2. The calculated is plotted as a function of 2 / 1 . It shows that the 2 = 0 gives a maximum value due to 1 ≫ 2 (i.e., the highly anisotropic SOC). To show the suppression of by SOC, we calculated as a function of 2 / 1 for 2 = 0. As shown in Fig. S14(b), we find that the increase of 2 / 1 suppresses the . Therefore, the effect of anisotropic SOC on the 2 irrep spin-triplet pairing is weak. This conclusion is general and can be directly applied to the (Nb,Ta)RuSi superconductor, where the DFT calculation could confirm 2 1 = 0.1.
(2) Energy compensation from spin magnetism According to our previous work (see Ref. [70] in the main text), the 1 + 2 can generate an out-of-plane spin magnetism via a third-order coupling term Im[Δ Δ * ]. As also discussed in the Note S1, the free energy of a homogeneous superconductor is given by = ( )|Δ | 2 + ( )|Δ | 2 + | ⃗⃗⃗ | 2 + 1 (Δ Δ ) 2 + 2 Δ Δ + . .. The minimization of free energy leads to = − 2 Im[Δ * Δ ]. More importantly, the coefficient 2 = 7 (3) 8 3 is proportional to the strength of spin-orbit coupling . Thus, this energy loss is due to the suppression of Δ by the spin-orbit coupling , while the energy gain is given by − 2 2 |Δ * Δ | 2 . Both these two effects are closely related to , the fully-gapped 1 + 2 pairing becomes possible if the spin magnetism term wins (i.e., the energy gain is larger than the energy loss). Moreover, this (s+ip) model shows a very good agreement with the superfluid density (see Fig. 3 in the main text).
Considering the above two reasons, we conclude that the non-symmetric (s+ip) pairing is the most consistent one with the (Nb,Ta)RuSi superconductors in the absence of disorder.